Snapping a vertex group to another
Destructive Method: Bridge

Duplicate the vertices of the left mesh X. Go to the target mesh, duplicate the target vertices (box select B and duplicate ⇧ ShiftD) and separate them P by selection. Join the target vertices with the original mesh.
Select the outer edges of the remaining target mesh, which are now unconnected (⎇ Alt RMB for edge loop selection). ⇧ Shift select the outer target vertices. Then bridge them with ⎈ CtrlE &gt; Bridge Edge Loops. Recalculate the normals of the target vertices. ⎈ CtrlN

Non-Destructive Method: Shapekeys

Select the vertices in question of the left original mesh. Assign them to a group ('Group'). Create the Basis shapekey and a new shapekey. In the new shapekey, move the original vertices close to the target mesh vertices. (As close as possible.)
Set the Shapekey influence value to 1.0. Add a Shrinkwrap modifier to the original object. Set the target object as Target and the Group as Vertex Group. Set the method to Nearest Vertex. The vertices now snap to the nearest vertex of the target mesh.
Apply the modifier as shapekey. You can delete the temporary shapekey.